Bayesian Logistic Regression for Predicting Diabetes Risk Using NHANES 2013–2014 Data

A Capstone Project on Bayesian Applications in Epidemiologic Modeling

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Introduction

Diabetes mellitus (DM) remains a major public health challenge, and identifying key risk factors—such as obesity, age, sex, and race/ethnicity—is essential for prevention and targeted intervention. Logistic regression is widely used to estimate associations between such factors and binary outcomes like diabetes diagnosis. However, classical maximum likelihood estimation (MLE) can produce unstable estimates in the presence of missing data, quasi-separation, or small samples. Bayesian logistic regression offers a robust alternative by integrating prior information, regularizing estimates, and quantifying uncertainty more transparently than frequentist approaches.

Bayesian hierarchical models, implemented via Markov Chain Monte Carlo (MCMC), have been successfully applied in predicting patient health status across diseases such as pneumonia, prostate cancer, and mental disorders (Zeger et al. 2020). By representing predictive uncertainty alongside point estimates, Bayesian inference offers a practical advantage in epidemiologic modeling where decisions hinge on probabilistic thresholds. Beyond stability, Bayesian methods support model checking, variable selection, and uncertainty quantification under missingness or imputation frameworks (Baldwin and Larson 2017; Kruschke and Liddell 2017).

Recent work has expanded Bayesian applications to disease diagnostics and health risk modeling. For instance, Bayesian approaches have been used to evaluate NHANES diagnostic data (Chatzimichail and Hatjimihail 2023), to model cardiovascular and metabolic risk (Liu et al. 2013), and to integrate multiple data modalities such as imaging and laboratory measures (Abdullah, Hassan, and Mustafa 2022). Moreover, multiple imputation combined with Bayesian modeling generates robust estimates when data are missing at random (MAR) or not at random (MNAR) (Austin et al. 2021).

The broader Bayesian literature emphasizes the role of priors and model checking. Weakly informative priors, such as \(N(0, 2.5)\) for coefficients, regularize estimation and reduce variance in small samples (Gelman et al. 2008; Vande Schoot et al. 2021). Tutorials using R packages like brms and blavaan illustrate how MCMC enables posterior inference and empirical Bayes analysis (Klauenberg et al. 2015).

Beyond standard generalized linear models, Bayesian nonparametric regression flexibly captures nonlinearity and zero inflation common in health data (Richardson and Hartman 2018). Bayesian Additive Regression Trees (BART) improve variable selection in mixed-type data (Luo et al. 2024), while state-space and dynamic Bayesian models incorporate time-varying biomarkers for longitudinal prediction (Momeni et al. 2021). Bayesian model averaging (BMA) further addresses model uncertainty by weighting across multiple specifications (Hoeting et al. 1999). Together, these approaches demonstrate the versatility and growing importance of Bayesian inference in clinical and epidemiologic modeling.

The objective of this project is to evaluate whether Bayesian inference provides more stable and interpretable estimates of diabetes risk than frequentist and imputation-based approaches, particularly when data complexity or separation challenges arise. Agreement across modeling frameworks supports the robustness of these associations and highlights the interpretability and uncertainty quantification advantages offered by Bayesian analysis in population health modeling (National Center for Health Statistics (NCHS) 2014).

Aims

The present study employs Bayesian logistic regression to predict diabetes status and examine the relationships between diabetes and key predictors, including body mass index (BMI), age (≥20 years), sex, and race. Using retrospective data from the 2013–2014 NHANES survey, the analysis accounts for the study’s complex sampling design, which involves stratification, clustering, and the oversampling of specific subpopulations rather than simple random sampling. The Bayesian framework is applied to address common analytical challenges such as missing data, complete case bias, and data separation, thereby improving the robustness and reliability of inference compared to traditional logistic regression methods.

Method

Bayesian Logistic Regression

The Bayesian framework integrates prior knowledge with observed data to generate posterior distributions, allowing parameters to be interpreted directly in probabilistic terms.

Unlike traditional frequentist approaches that yield single-point estimates and p-values, Bayesian methods represent parameters as random variables with full probability distributions.

This provides greater flexibility, incorporates parameter uncertainty, and produces credible intervals that directly quantify the probability that a parameter lies within a given range.

Model Structure

Bayesian logistic regression models the log-odds of a binary outcome as a linear combination of predictors:

\[ \text{logit}(P(Y = 1)) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k \]

where

• \(P(Y = 1)\) is the probability of the event of interest,
• \(\beta_0\) is the intercept (log-odds when all predictors are zero), and
• \(\beta_j\) represents the effect of predictor \(X_j\) on the log-odds of the outcome, holding other predictors constant.

In the Bayesian framework, model parameters (\(\boldsymbol{\beta}\)) are treated as random variables and assigned prior distributions that reflect existing knowledge or plausible ranges before observing the data. After incorporating the observed evidence, the priors are updated through Bayes’ theorem (Leeuw and Klugkist 2012; Klauenberg et al. 2015):

\[ \text{Posterior} \propto \text{Likelihood} \times \text{Prior} \]

• Likelihood: represents the probability of the observed data given the model parameters—it captures how well different parameter values explain the data.
• Prior: expresses beliefs or existing information about the parameters before observing the data.
• Posterior: combines both, representing the updated distribution of parameter values after accounting for the data.

This formulation allows uncertainty to propagate naturally through the model, producing posterior distributions for each coefficient that can be directly interpreted as probabilities.

Prior Specification

Weakly informative priors were used to regularize estimation without imposing strong assumptions:

• Regression coefficients: \(N(0, 2.5)\), providing gentle regularization while allowing substantial variation in plausible effects (Gelman et al. 2008; Vande Schoot et al. 2021).
• Intercept: Student’s t-distribution prior, \(t(3, 0, 10)\) (Schoot et al. 2013; Vande Schoot et al. 2021), which has
  • 3 degrees of freedom (heavy tails to allow occasional large effects),
  • mean 0 (no bias toward positive or negative effects), and
  • scale 10 (broad range of possible values).

Such priors help stabilize estimation in the presence of multicollinearity, limited sample size, or potential outliers.

Advantages of Bayesian Logistic Regression

• Uncertainty quantification: Produces full posterior distributions instead of single estimates.
• Credible intervals: Provide the range within which a parameter lies with a specified probability (e.g., 95%).
• Flexible priors: Allow integration of expert knowledge or findings from prior studies.
• Probabilistic predictions: Posterior predictive distributions yield direct probabilities for new or future observations.
• Model evaluation: Posterior predictive checks (PPCs) assess how well simulated outcomes reproduce observed data.

Posterior Predictions

Posterior distributions of regression coefficients were used to estimate the probability of the outcome for given predictor values. This allows statements such as:

Given the predictors, the probability of the outcome lies between X% and Y%.

Posterior predictions account for two key sources of uncertainty:

1. Parameter uncertainty: Variability in estimated model coefficients.
2. Predictive uncertainty: Variability in possible future outcomes given those parameters.

In Bayesian analysis, all unknown quantities—coefficients, means, variances, or probabilities—are treated as random variables described by their posterior distributions.

Model Evaluation and Diagnostics

Model quality and convergence were assessed using standard Bayesian diagnostics:

• Posterior sampling: Conducted via Markov Chain Monte Carlo (MCMC) using the No-U-Turn Sampler (NUTS), a variant of Hamiltonian Monte Carlo (HMC) (Austin et al. 2021). Four chains were run with sufficient warm-up iterations to ensure convergence.
• Convergence metrics: The potential scale reduction factor (\(\hat{R}\)) and effective sample size (ESS) were used to verify stability and mixing across chains.
• Autocorrelation checks: Ensured independence between successive draws.
• Posterior predictive checks (PPCs): Compared simulated outcomes to observed data to evaluate fit.
• Bayesian \(R^2\): Quantified the proportion of variance explained by predictors, incorporating posterior uncertainty.

Analysis and Results

Data Preparation

This study used publicly available 2013–2014 NHANES data published by the CDC’s National Center for Health Statistics (National Center for Health Statistics (NCHS) 2014). Three component files were utilized: DEMO_H (demographics), BMX_H (body measures), and DIQ_H (diabetes questionnaire). Each file was imported in .XPT format using the haven package in R, and merged using the unique participant identifier SEQN to create a single adult analytic dataset (age ≥ 20 years).

All variables were coerced to consistent numeric or factor types prior to merging to ensure atomic columns suitable for survey-weighted analysis and modeling. The use of SEQN preserved respondent integrity across datasets and ensured accurate record linkage. This preprocessing step standardized variable formats and minimized inconsistencies between files.

Data wrangling, cleaning, and merging were performed in R using a combination of base functions and tidyverse packages. Bayesian logistic regression modeling was later implemented using the brms interface to Stan, allowing probabilistic inference within a reproducible workflow that accommodated the NHANES complex survey design and missing data considerations.

Data Import and Merging

merged_data <- readRDS("data/merged_2013_2014.rds")

merged_n <- nrow(merged_data)

The merged dataset contains 10,175 participants. It integrates demographic, examination, and diabetes questionnaire data. We then restrict the sample to adults (age ≥ 20) to define the analytic cohort used in subsequent analyses. A small proportion of records have missing values in BMI and diabetes status, which will be addressed later through multiple imputation.

Preview of merged NHANES 2013–2014 dataset limited to analysis variables (source columns only).

RIDAGEYR	BMXBMI	RIAGENDR	RIDRETH1	DIQ010
69	26.7	1	4	1
54	28.6	1	3	1
72	28.9	1	3	1
9	17.1	1	3	2
73	19.7	2	3	2
56	41.7	1	1	2
0	NA	1	3	NA
61	35.7	2	3	2
42	NA	1	2	2
56	26.5	2	3	2

Variable Definitions

• Response Variable:
  diabetes_ind (binary) represents a Type 2 diabetes diagnosis, excluding gestational diabetes. It was derived from DIQ010 (“Doctor told you have diabetes”), while DIQ050 (insulin use) was excluded to prevent treatment-related confounding.

• Predictor Variables:

  • BMXBMI – Body Mass Index (categorical; six levels as bmi_cat)
    
  • RIDAGEYR – Age (continuous, 20–80 years)
    
  • RIAGENDR – Sex (factor, two levels)
    
  • RIDRETH1 – Ethnicity (factor, five levels)

library(knitr)
library(kableExtra)
library(dplyr)
library(tibble)

# -----------------------------
# Variable descriptions (with `code` formatting for names)
# -----------------------------
var_tbl <- tribble(
  ~Variable,      ~Description,                                                                                                   ~Type,         ~Origin,
  "`diabetes_ind`","Type 2 diabetes diagnosis (1 = Yes, 0 = No) derived from `DIQ010`; gestational diabetes excluded.",           "Categorical", "Derived from `DIQ010`",
  "`age`",        "Age in years.",                                                                                                "Continuous",  "NHANES `RIDAGEYR`",
  "`bmi`",        "Body Mass Index (kg/m^2) computed from measured height and weight.",                                           "Continuous",  "NHANES `BMXBMI`",
  "`bmi_cat`",    "BMI categories: Underweight, Normal, Overweight, Obesity I–III (`Normal` is reference in models).",            "Categorical", "Derived from `bmi`",
  "`sex`",        "Sex of participant (`Male`, `Female`).",                                                                       "Categorical", "NHANES `RIAGENDR`",
  "`race`",       "Race/Ethnicity with five levels (e.g., Non-Hispanic White, Non-Hispanic Black, Mexican American).",            "Categorical", "NHANES `RIDRETH1`",
  "`WTMEC2YR`",   "Examination sample weight for Mobile Examination Center participants.",                                        "Weight",      "NHANES design",
  "`SDMVPSU`",    "Primary Sampling Unit used for variance estimation in the complex survey design.",                             "Design",      "NHANES design",
  "`SDMVSTRA`",   "Stratum identifier used to define strata for the complex survey design.",                                      "Design",      "NHANES design",
  "`age_c`",      "Centered and standardized age (z-score).",                                                                     "Continuous",  "Derived from `age`",
  "`bmi_c`",      "Centered and standardized BMI (z-score).",                                                                     "Continuous",  "Derived from `bmi`"
)

kbl(
  var_tbl,
  caption = "Variable Descriptions: Adult Analytic Dataset",
  align = c("l","l","l","l"),
  escape = FALSE
) %>%
  kable_styling(full_width = FALSE, position = "center", bootstrap_options = c("striped","hover")) %>%
  group_rows("Analysis variables", 1, 6) %>%              # <-- updated range (now 6 analysis rows)
  group_rows("Survey design variables", 7, 9) %>%
  group_rows("Derived variables", 10, 11)

Variable Descriptions: Adult Analytic Dataset

Variable	Description	Type	Origin
Analysis variables
`diabetes_ind`	Type 2 diabetes diagnosis (1 = Yes, 0 = No) derived from `DIQ010`; gestational diabetes excluded.	Categorical	Derived from `DIQ010`
`age`	Age in years.	Continuous	NHANES `RIDAGEYR`
`bmi`	Body Mass Index (kg/m^2) computed from measured height and weight.	Continuous	NHANES `BMXBMI`
`bmi_cat`	BMI categories: Underweight, Normal, Overweight, Obesity I–III (`Normal` is reference in models).	Categorical	Derived from `bmi`
`sex`	Sex of participant (`Male`, `Female`).	Categorical	NHANES `RIAGENDR`
`race`	Race/Ethnicity with five levels (e.g., Non-Hispanic White, Non-Hispanic Black, Mexican American).	Categorical	NHANES `RIDRETH1`
Survey design variables
`WTMEC2YR`	Examination sample weight for Mobile Examination Center participants.	Weight	NHANES design
`SDMVPSU`	Primary Sampling Unit used for variance estimation in the complex survey design.	Design	NHANES design
`SDMVSTRA`	Stratum identifier used to define strata for the complex survey design.	Design	NHANES design
Derived variables
`age_c`	Centered and standardized age (z-score).	Continuous	Derived from `age`
`bmi_c`	Centered and standardized BMI (z-score).	Continuous	Derived from `bmi`

Study Design and Survey-Weighted Analysis

The National Health and Nutrition Examination Survey (NHANES) employs a complex, multistage probability sampling design with stratification, clustering, and oversampling of specific demographic groups (for example, racial/ethnic minorities and older adults) to produce nationally representative estimates of the U.S. population.

Survey design variables — primary sampling units (SDMVPSU), strata (SDMVSTRA), and examination sample weights (WTMEC2YR) — were retained to account for this complex design. These variables were applied to adjust for unequal probabilities of selection, nonresponse, and oversampling, ensuring valid standard errors, unbiased prevalence estimates, and generalizable population-level inference.

A survey-weighted logistic regression model was used to evaluate the association between diabetes status (diabetes_ind, binary outcome) and key predictors: body mass index (bmi), age (age), sex (sex), and race/ethnicity (race). Diabetes was defined using DIQ010 (“Doctor told you have diabetes”) and coded as 0/1, with DIQ050 (insulin use) excluded to avoid treatment-related confounding.

Covariates included:
- age (continuous; centered as age_c, categorized 20–80 years)
- bmi (continuous; centered as bmi_c, and categorized by BMI class bmi_cat)
- sex (male, female)
- race (five ethnicity levels)

This approach accounts for NHANES’ complex sampling design, producing unbiased parameter estimates and valid inference for U.S. adults.

Step	Description
Weighting	Used the survey package to calculate weighted means and standard deviations for all variables.
Standardization	Centered and standardized BMI and age (bmi_c, age_c) for use in regression models.
Age Categorization	Recoded into intervals: 20–<30, 30–<40, 40–<50, 50–<60, 60–<70, and 70–80 years.
BMI Categorization	Recoded as: <18.5 (Underweight), 18.5–<25 (Normal), 25–<30 (Overweight), 30–<35 (Obesity I), 35–<40 (Obesity II), ≥40 (Obesity III).
Ethnicity Recoding	Recoded as: 1 = Mexican American, 2 = Other Hispanic, 3 = Non-Hispanic White, 4 = Non-Hispanic Black, 5 = Other/Multi.
Special Codes	Transformed nonresponse codes (e.g., 3, 7) to NA. These missing codes were evaluated for potential nonrandom patterns (MAR/MNAR).
Missing Data	Retained and visualized missing values to assess their pattern and informativeness before multiple imputation.
Final Dataset	Created the cleaned analytic dataset (adult) using Non-Hispanic White and Male as reference groups for modeling.

Adult Cohort Definition

library(survey)

# NHANES survey design object for the adult analytic cohort

nhanes_design_adult <- survey::svydesign(
id      = ~SDMVPSU,
strata  = ~SDMVSTRA,
weights = ~WTMEC2YR,
nest    = TRUE,
data    = adult
)

# Quick weighted checks

survey::svymean(~age, nhanes_design_adult, na.rm = TRUE)

      mean     SE
age 47.496 0.3805

survey::svymean(~diabetes_ind, nhanes_design_adult, na.rm = TRUE)

                mean     SE
diabetes_ind 0.10391 0.0052

# Design effect and effective sample size for `diabetes_ind`

v_hat <- as.numeric(survey::svyvar(~diabetes_ind, nhanes_design_adult, na.rm = TRUE))
p_hat <- mean(adult$diabetes_ind, na.rm = TRUE)
n_obs <- nrow(adult)
v_srs <- p_hat * (1 - p_hat) / n_obs
deff  <- v_hat / v_srs

n_total <- sum(weights(nhanes_design_adult), na.rm = TRUE)
ess     <- as.numeric(n_total / deff)

cat("Design effect for diabetes_ind:", round(deff, 2), "\n")

Design effect for diabetes_ind: 4777.96 

cat("Effective sample size for diabetes_ind:", round(ess), "\n")

Effective sample size for diabetes_ind: 47960 

Descriptive statistics for continuous and categorical variables are presented below.

# Keep only analytic variables for Table 1
tbl1_dat <- adult %>%
  transmute(
    age,
    bmi,
    bmi_cat,
    sex,
    race4,
    # CHANGE: make diabetes_ind a factor so it plays nice with pivot_longer()
    diabetes_ind = factor(diabetes_ind, levels = c(0, 1), labels = c("No", "Yes"))
  )

# Continuous summaries: n, missing, mean, sd, min, max
cont_vars <- c("age", "bmi")

cont_sum <- tbl1_dat %>%
  select(all_of(cont_vars)) %>%
  pivot_longer(everything(), names_to = "Variable", values_to = "value") %>%
  group_by(Variable) %>%
  summarise(
    N          = sum(!is.na(value)),
    Missing    = sum(is.na(value)),
    Mean       = mean(value, na.rm = TRUE),
    SD         = sd(value, na.rm = TRUE),
    Min        = min(value, na.rm = TRUE),
    Max        = max(value, na.rm = TRUE),
    .groups = "drop"
  ) %>%
  mutate(across(c(Mean, SD, Min, Max), ~round(.x, 2)))

# Categorical summaries: counts and percents
cat_vars <- c("sex", "race4", "diabetes_ind", "bmi_cat")

cat_sum <- tbl1_dat %>%
  # CHANGE: ensure all cat vars are factors and include NA as an explicit "(Missing)" level
  mutate(across(all_of(cat_vars),
                ~ forcats::fct_explicit_na(as.factor(.x), na_level = "(Missing)"))) %>%
  select(all_of(cat_vars)) %>%
  pivot_longer(everything(), names_to = "Variable", values_to = "Level") %>%
  count(Variable, Level, name = "n") %>%
  group_by(Variable) %>%
  mutate(pct = round(100 * n / sum(n), 1)) %>%
  ungroup() %>%
  arrange(Variable, desc(n))

kable(cont_sum, caption = "Table 1a. Continuous variables (age, BMI): N, missing, mean (SD), range.")

Table 1a. Continuous variables (age, BMI): N, missing, mean (SD), range.

Variable	N	Missing	Mean	SD	Min	Max
age	5769	0	49.11	17.56	20.0	80.0
bmi	5520	249	29.10	7.15	14.1	82.9

kable(cat_sum,  caption = "Table 1b. Categorical variables (sex, race4, diabetes_ind, bmi_cat): counts and percentages.")

Table 1b. Categorical variables (sex, race4, diabetes_ind, bmi_cat): counts and percentages.

Variable	Level	n	pct
bmi_cat	Overweight	1768	30.6
bmi_cat	Normal	1579	27.4
bmi_cat	Obesity I	1145	19.8
bmi_cat	Obesity II	519	9.0
bmi_cat	Obesity III	419	7.3
bmi_cat	(Missing)	249	4.3
bmi_cat	Underweight	90	1.6
diabetes_ind	No	4870	84.4
diabetes_ind	Yes	722	12.5
diabetes_ind	(Missing)	177	3.1
race4	White	2472	42.8
race4	Hispanic	1275	22.1
race4	Black	1177	20.4
race4	Other	845	14.6
sex	Female	3011	52.2
sex	Male	2758	47.8

Table 1a and 1b summarize the analytic variables included in subsequent models. Mean age and BMI values indicate an adult cohort spanning a wide range of body composition, while categorical summaries confirm balanced sex representation and sufficient sample sizes across race/ethnicity categories. These variables were standardized and used as predictors in all modeling frameworks.

adult_n <- nrow(adult)

Table 1: Excerpt of the NHANES 2013–2014 adult cohort (age ≥ 20; N = 5,769) with derived and standardized variables.

SDMVPSU	SDMVSTRA	WTMEC2YR	diabetes_ind	bmi	age	sex	race	age_c	bmi_c	bmi_cat	race4
1	112	13481.04	1	26.7	69	Male	NH Black	1.1324183	-0.3358861	Overweight	Black
1	108	24471.77	1	28.6	54	Male	NH White	0.2783598	-0.0702810	Overweight	White
1	109	57193.29	1	28.9	72	Male	NH White	1.3032300	-0.0283434	Overweight	White
2	116	65541.87	0	19.7	73	Female	NH White	1.3601672	-1.3144311	Normal	White
1	111	25344.99	0	41.7	56	Male	Mexican American	0.3922343	1.7609961	Obesity III	Hispanic
1	114	61758.65	0	35.7	61	Female	NH White	0.6769204	0.9222432	Obesity II	White

As shown in Table 1, the analytic adult cohort (N = 5,769) includes standardized variables for age and BMI (age_c, bmi_c), categorical indicators for sex and race/ethnicity (race4), and a binary doctor-diagnosed diabetes variable (diabetes_ind).

# Textual structure and preview
str(adult)

'data.frame':   5769 obs. of  12 variables:
 $ SDMVPSU     : num  1 1 1 2 1 1 2 1 2 2 ...
 $ SDMVSTRA    : num  112 108 109 116 111 114 106 112 112 113 ...
 $ WTMEC2YR    : num  13481 24472 57193 65542 25345 ...
 $ diabetes_ind: num  1 1 1 0 0 0 0 0 0 0 ...
 $ bmi         : num  26.7 28.6 28.9 19.7 41.7 35.7 NA 26.5 22 20.3 ...
 $ age         : num  69 54 72 73 56 61 42 56 65 26 ...
 $ sex         : Factor w/ 2 levels "Female","Male": 2 2 2 1 2 1 2 1 2 1 ...
 $ race        : Factor w/ 5 levels "Mexican American",..: 2 3 3 3 1 3 4 3 3 3 ...
 $ age_c       : num  1.132 0.278 1.303 1.36 0.392 ...
 $ bmi_c       : num  -0.3359 -0.0703 -0.0283 -1.3144 1.761 ...
 $ bmi_cat     : Factor w/ 7 levels "Underweight",..: 3 3 3 2 6 5 7 3 2 2 ...
 $ race4       : Factor w/ 4 levels "White","Hispanic",..: 3 1 1 1 2 1 2 1 1 1 ...

# Visual structure and type overview
plot_intro(adult, title = "Adult dataset: variable types and completeness")

library(DataExplorer)

# Visualize missing data pattern for key variables

plot_missing(adult, title = "Missing data pattern (Adult dataset)")

Missingness overview for the adult dataset.

miss_tbl <- tibble::tibble(
Variable = c("`bmi`","`diabetes_ind`"),
Missing_n = c(sum(is.na(adult_eda$bmi)),
sum(is.na(adult_eda$diabetes_ind))),
Missing_pct = round(c(mean(is.na(adult_eda$bmi)),
mean(is.na(adult_eda$diabetes_ind))) * 100, 1)
)

knitr::kable(
miss_tbl,
caption = "Missingness for key analysis variables."
)

Missingness for key analysis variables.

Variable	Missing_n	Missing_pct
bmi	249	4.3
diabetes_ind	177	3.1

Missing Data Summary

Missingness was low overall (~7.3%). Gaps were concentrated in bmi and diabetes_ind; design variables and demographics were complete. Patterns were consistent with MAR, so we used multiple imputation (MICE) prior to modeling.

# Summarize missingness by variable

colSums(is.na(adult_eda))

     SDMVPSU     SDMVSTRA     WTMEC2YR diabetes_ind          bmi          age 
           0            0            0          177          249            0 
         sex         race        age_c        bmi_c      bmi_cat        race4 
           0            0            0          249            0            0 

# Focus on expected variables with NAs

sapply(adult_eda[, c("diabetes_ind", "bmi", "bmi_c", "age")], function(x) sum(is.na(x)))

diabetes_ind          bmi        bmi_c          age 
         177          249          249            0 

# Check for missing or invalid source codes

table(merged_data$DIQ010, useNA = "ifany")

   1    2    3    7    9 <NA> 
 737 8841  185    1    5  406 

sum(is.na(merged_data$BMXBMI))

[1] 1120

# (Note: full missingness pattern already visualized above using plot_missing(adult).)

Results indicated that missing values were limited primarily to BMI and diabetes indicators, while demographic and survey design variables were complete.

The overall missingness (~4%) was low and appears consistent with data missing at random (MAR). This pattern likely reflects participation differences in the physical examination component among older adults or individuals with health limitations.

Given these findings, multiple imputation using chained equations (MICE) was employed to minimize bias and retain statistical power in subsequent modeling.

Exploratory Data Summary

Following the missing data assessment, exploratory analyses were conducted to describe the adult analytic cohort and visualize distributions across key demographic and health variables. The goal was to examine univariate patterns and bivariate relationships relevant to diabetes prevalence before modeling.

The adult analytic cohort was broadly representative of the U.S. population, with a majority identifying as Non-Hispanic White. Age and BMI distributions were right-skewed, with most participants classified as overweight or obese. Visual exploration revealed a clear positive relationship between age, BMI, and diabetes prevalence. Non-Hispanic Black and Hispanic participants exhibited higher proportions of diabetes compared to Non-Hispanic Whites.

Approximately 25% of variables were categorical (for example, sex, race4, diabetes_ind) and 75% were continuous (age, bmi, age_c, bmi_c), indicating that the dataset primarily consisted of measured numeric values such as BMI and age. About 93% of rows contained complete information across all predictors and outcomes, reflecting high data quality.

Age was relatively evenly distributed across adult age groups, while BMI was concentrated in the overweight and obese ranges. Female participants were slightly overrepresented relative to Male participants.

# Age distribution (analytic adult)
ggplot(adult, aes(x = age)) +
  geom_histogram(binwidth = 5, color = "white") +
  labs(title = "Distribution of Age (≥20 years)", x = "Age (years)", y = "Count") +
  theme_minimal()

# Diabetes outcome distribution
ggplot(adult, aes(x = factor(diabetes_ind, levels = c(0,1), labels = c("No","Yes")))) +
  geom_bar() +
  labs(title = "Diabetes Outcome Distribution (≥20 years)", x = "Diabetes (No/Yes)", y = "Count") +
  theme_minimal()

# BMI category distribution
ggplot(adult, aes(x = bmi_cat)) +
  geom_bar(color = "white", fill = "skyblue") +
  labs(title = "Distribution of BMI Categories (≥20 years)", x = "BMI Category", y = "Count") +
  theme_minimal()

# BMI by diabetes outcome (boxplot)
# (You can’t use boxplot with categorical y, so revert to numeric BMI here)
ggplot(adult, aes(x = factor(diabetes_ind, levels = c(0,1), labels = c("No","Yes")), y = bmi)) +
  geom_boxplot(fill = "lightblue") +
  labs(title = "BMI by Diabetes Diagnosis (≥20 years)", x = "Diabetes (No/Yes)", y = "BMI (numeric)") +
  theme_minimal()

# Diabetes by race4 (dodged bars)
ggplot(adult, aes(x = race4, fill = factor(diabetes_ind, levels = c(0,1), labels = c("No","Yes")))) +
  geom_bar(position = "dodge") +
  labs(title = "Diabetes Diagnosis by Race/Ethnicity (≥20 years)",
       x = "Race/Ethnicity (race4)", y = "Count", fill = "Diabetes") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

The EDA missingness summary shows approximately 4.3% missing BMI and 3.1% missing diabetes status (diabetes_ind). All design variables (WTMEC2YR, SDMVPSU, SDMVSTRA), as well as age, sex, and race4, are complete—sex and race NAs are encoded as explicit “(Missing)” levels in the EDA view.

Modeling Frameworks

Three modeling frameworks were compared using identical predictors (standardized age, BMI, sex, and race4) and the binary outcome diabetes_ind: (1) survey-weighted logistic regression to incorporate the NHANES complex sampling design, (2) multiple imputation (MICE) to address missing BMI values, and (3) Bayesian logistic regression with weakly informative priors to quantify uncertainty.

Survey-Weighted Logistic Regression (Design-Based MLE)

adult_clean <- adult %>%
  dplyr::mutate(
    sex   = forcats::fct_drop(sex),
    race4 = forcats::fct_drop(race4),
    age_c = as.numeric(age_c),
    bmi_c = as.numeric(bmi_c)
  ) %>%
  dplyr::filter(!is.na(diabetes_ind), !is.na(age_c), !is.na(bmi_c)) %>%
  droplevels()

str(adult_clean[, c("diabetes_ind","sex","race4","age_c","bmi_c")])

'data.frame':   5349 obs. of  5 variables:
 $ diabetes_ind: num  1 1 1 0 0 0 0 0 0 1 ...
 $ sex         : Factor w/ 2 levels "Female","Male": 2 2 2 1 2 1 1 2 1 2 ...
 $ race4       : Factor w/ 4 levels "White","Hispanic",..: 3 1 1 1 2 1 1 1 1 1 ...
 $ age_c       : num  1.132 0.278 1.303 1.36 0.392 ...
 $ bmi_c       : num  -0.3359 -0.0703 -0.0283 -1.3144 1.761 ...

options(survey.lonely.psu = "adjust")

nhanes_design_adult <- survey::svydesign(
id = ~SDMVPSU,
strata = ~SDMVSTRA,
weights = ~WTMEC2YR,
nest = TRUE,
data = adult_clean
)

des_cc <- subset(nhanes_design_adult, !is.na(diabetes_ind) &
!is.na(age_c) & !is.na(bmi_c) &
!is.na(sex) & !is.na(race4))

# Sanity checks before modeling
stopifnot(nlevels(adult_clean$sex)   >= 2)
stopifnot(nlevels(adult_clean$race4) >= 2)

table(adult_clean$sex)

Female   Male 
  2798   2551 

table(adult_clean$race4)

   White Hispanic    Black    Other 
    2293     1183     1101      772 

svy_fit <- survey::svyglm(
diabetes_ind ~ age_c + bmi_c + sex + race4,
design = des_cc,
family = quasibinomial()
)

svy_or <- broom::tidy(svy_fit, conf.int = TRUE) %>%
dplyr::mutate(
OR = exp(estimate),
LCL = exp(conf.low),
UCL = exp(conf.high)
) %>%
dplyr::select(term, OR, LCL, UCL, p.value) %>%
dplyr::filter(term != "(Intercept)")

Design-based odds ratios are summarized in Table 2.

knitr::kable(svy_or)

Table 2: Survey-weighted logistic regression: odds ratios (OR) and 95% confidence intervals for diabetes diagnosis among adults (NHANES 2013–2014).

term	OR	LCL	UCL	p.value
age_c	2.977668	2.704677	3.278212	0.0000000
bmi_c	1.930284	1.679190	2.218924	0.0000021
sexMale	1.287236	1.018664	1.626618	0.0373081
race4Hispanic	1.809943	1.428957	2.292507	0.0003024
race4Black	1.599844	1.157393	2.211434	0.0094754
race4Other	2.195356	1.461219	3.298334	0.0017976

The NHANES 2013–2014 data use a complex, multistage probability design involving strata (SDMVSTRA), primary sampling units (PSUs; SDMVPSU), and examination weights (WTMEC2YR) to ensure nationally representative estimates (National Center for Health Statistics (NCHS) 2014).

Estimates are population-weighted using NHANES survey design variables (WTMEC2YR, SDMVSTRA, SDMVPSU). Odds ratios are reported per one standard deviation (1 SD) increase in age and BMI, with reference groups Male and White.

Multiple Imputation (MICE)

Multiple Imputation by Chained Equations (MICE) was used as a principled approach for handling missing data (Stef van Buuren and Groothuis-Oudshoorn 2011; S. van Buuren 2012).
MICE iteratively imputes each incomplete variable using regression models based on other variables in the dataset, producing multiple completed datasets that reflect uncertainty due to missingness. Estimates are then pooled across imputations using Rubin’s rules to generate final parameter estimates and confidence intervals.

MICE, as an alternative to the Bayesian approach, effectively manages missing data through chained regression equations without requiring full joint modeling of all variables.

For large sample sizes (n ≥ 400), even in the presence of high percentages (up to 75%) of missing data in one variable, non-normal distributions such as flat densities, heavy tails, skewness, and multimodality do not materially affect mean structure estimation performance (S. van Buuren 2012).

In this study, continuous variables (age and BMI) were imputed using predictive mean matching (PMM) to preserve realistic distributions, while categorical variables (sex and race4) were imputed using logistic and polytomous regression models, respectively. Diabetes status (diabetes_ind) was treated as an outcome variable and was not imputed. Twenty imputations were generated to reduce Monte Carlo error and maintain robust variance estimation.

adult_imp1 <- mice::complete(imp, 1) %>%
  dplyr::mutate(
    sex     = factor(sex,   levels = levels(adult$sex)),
    race4   = factor(race4, levels = levels(adult$race4)),
    age_c   = as.numeric(scale(age)),
    bmi_c   = as.numeric(scale(bmi)),
    wt_norm = WTMEC2YR / mean(WTMEC2YR, na.rm = TRUE)
  ) %>%
  dplyr::filter(!is.na(diabetes_ind), !is.na(age_c), !is.na(bmi_c)) %>%
  droplevels()

fml_bayes <- diabetes_ind | weights(wt_norm) ~ age_c + bmi_c + sex + race4

priors <- c(
  brms::set_prior("normal(0, 2.5)", class = "b"),
  brms::set_prior("student_t(3, 0, 10)", class = "Intercept")
)

invisible(capture.output({
  bayes_fit <- brms::brm(
    formula = fml_bayes,
    data    = adult_imp1,
    family  = bernoulli(link = "logit"),
    prior   = priors,
    chains  = 4, iter = 2000, seed = 123,
    control = list(adapt_delta = 0.95),
    refresh = 0
  )
}))

bayes_or <- brms::posterior_summary(bayes_fit, pars = "^b_") %>%
  as.data.frame() %>%
  tibble::rownames_to_column("raw") %>%
  dplyr::mutate(
    term = gsub("^b_", "", raw),
    OR   = exp(Estimate),
    LCL  = exp(Q2.5),
    UCL  = exp(Q97.5)
  ) %>%
  dplyr::select(term, OR, LCL, UCL)

knitr::kable(
  dplyr::mutate(bayes_or, dplyr::across(c(OR, LCL, UCL), ~ round(.x, 2)))
)

Table 3: Bayesian logistic regression: posterior odds ratios (OR) with 95% credible intervals.

term	OR	LCL	UCL
Intercept	0.06	0.05	0.07
age_c	2.93	2.62	3.30
bmi_c	1.92	1.76	2.10
sexMale	1.28	1.06	1.55
race4Hispanic	1.82	1.40	2.38
race4Black	1.62	1.23	2.12
race4Other	2.11	1.46	2.99

Multiple imputation preserves sample size and reduces bias from missing BMI values. Results closely mirror the survey-weighted model, confirming robustness to imputation.

miss_age  <- sum(is.na(mi_dat$age))
miss_bmiN <- sum(is.na(mi_dat$bmi))

mi_caption <- if (miss_age > 0 && miss_bmiN > 0) {
  "Multiple Imputation (MICE): pooled odds ratios (OR) and 95% confidence intervals after imputing missing age (normal) and BMI (PMM) (m = 20); diabetes status was not imputed."
} else if (miss_bmiN > 0) {
  "Multiple Imputation (MICE): pooled odds ratios (OR) and 95% confidence intervals after imputing missing BMI (PMM) (m = 20); diabetes status was not imputed."
} else if (miss_age > 0) {
  "Multiple Imputation (MICE): pooled odds ratios (OR) and 95% confidence intervals after imputing missing age (normal) (m = 20); diabetes status was not imputed."
} else {
  "Multiple Imputation (MICE): pooled odds ratios (OR) and 95% confidence intervals (no variables required imputation); diabetes status was not imputed."
}
mi_caption <- paste0(mi_caption, " Odds ratios are per 1 SD for age and BMI.")

knitr::kable(mi_or, caption = mi_caption)

Table 4: Multiple Imputation (MICE): pooled odds ratios (OR) and 95% confidence intervals after imputing missing BMI (PMM) (m = 20); diabetes status was not imputed. Odds ratios are per 1 SD for age and BMI.

	term	OR	std.error	statistic	df	p.value	LCL	UCL	conf.low	conf.high
2	scale(age)	2.8956646	0.0524433	20.273592	5499.574	0.0000000	2.6127544	3.2092084	2.6127544	3.2092084
3	scale(bmi)	1.8053391	0.0430294	13.728961	3877.226	0.0000000	1.6592839	1.9642506	1.6592839	1.9642506
4	relevel(sex, “Male”)Female	0.8056102	0.0872427	-2.477631	5545.705	0.0132553	0.6789653	0.9558776	0.6789653	0.9558776
5	relevel(race4, “White”)Hispanic	2.0741944	0.1128115	6.467183	5562.036	0.0000000	1.6626591	2.5875915	1.6626591	2.5875915
6	relevel(race4, “White”)Black	1.7931172	0.1137153	5.135240	5508.172	0.0000003	1.4348045	2.2409110	1.4348045	2.2409110
7	relevel(race4, “White”)Other	2.0011166	0.1443011	4.807344	5464.535	0.0000016	1.5080503	2.6553940	1.5080503	2.6553940

Bayesian Logistic Regression

Bayesian logistic regression was implemented using the following model specification:

Formula:
diabetes_ind | weights(wt_norm) ~ age_c + bmi_c + sex + race4

stopifnot(all(c("diabetes_ind","age","bmi") %in% names(adult_imp1)))

correlation_matrix <- cor(
  adult_imp1[, c("diabetes_ind", "age", "bmi")],
  use = "complete.obs",
  method = "pearson"
)
correlation_melted <- melt(correlation_matrix, varnames = c("Var1","Var2"), value.name = "value")

ggplot(correlation_melted, aes(Var1, Var2, fill = value)) +
  geom_tile(color = "white") +
  scale_fill_gradient2(low = "blue", high = "red", mid = "white",
                       midpoint = 0, limits = c(-1, 1), name = "Correlation") +
  theme_minimal() +
  labs(title = "Correlation Heatmap (adult_imp1)", x = "Features", y = "Features")

bayes_fit_prior <- tryCatch(
  update(bayes_fit, sample_prior = "yes", refresh = 0),
  error = function(e) NULL
)

Running MCMC with 4 sequential chains...

Chain 1 finished in 10.9 seconds.
Chain 2 finished in 10.2 seconds.
Chain 3 finished in 11.1 seconds.
Chain 4 finished in 10.6 seconds.

All 4 chains finished successfully.
Mean chain execution time: 10.7 seconds.
Total execution time: 43.2 seconds.

Posterior odds ratios and credible intervals from the Bayesian logistic regression are shown in Table 3.

As shown in Table 3, the Bayesian logistic regression model estimated the log-odds of diabetes using standardized predictors. Weakly informative priors (\(N(0, 2.5)\) for slopes, Student-t(3, 0, 10) for the intercept) stabilized estimation and prevented overfitting. The model used normalized NHANES exam weights as importance weights to approximate design-based inference. Posterior means and 95% credible intervals provided full uncertainty quantification for each predictor.

Posterior summaries were further evaluated using the Bayesian \(R^2\), which estimates the proportion of outcome variance explained by model predictors.

Model-level performance is summarized in Table 5.

knitr::kable(as.data.frame(brms::bayes_R2(bayes_fit)))

Table 5: Bayesian R² summary.

	Estimate	Est.Error	Q2.5	Q97.5
R2	0.1380082	0.0118579	0.115269	0.1616909

# Get diagnostics as plain columns (use metric names as strings)
diag <- posterior::summarise_draws(
  bayes_fit,
  "rhat", "ess_bulk", "ess_tail"
)

# Keep only slope parameters (b_*) and build a clean table.
diag_b <- diag |>
  dplyr::as_tibble() |>
  dplyr::filter(grepl("^b_", .data$variable)) |>
  dplyr::transmute(
    Parameter = .data$variable,
    Rhat      = .data$rhat,
    Bulk_ESS  = .data$ess_bulk,
    Tail_ESS  = .data$ess_tail
  )

knitr::kable(diag_b, digits = 1)

Table 6: MCMC diagnostics (R-hat and Effective Sample Sizes) for model parameters (including intercept).

Parameter	Rhat	Bulk_ESS	Tail_ESS
b_Intercept	1	2721.2	2817.8
b_age_c	1	2605.7	2704.7
b_bmi_c	1	3229.5	2783.0
b_sexMale	1	3613.7	3128.7
b_race4Hispanic	1	3805.4	3109.7
b_race4Black	1	3685.6	3133.0
b_race4Other	1	3513.0	2637.5

All model parameters achieved R̂ values approximately equal to 1.00 and bulk/tail effective sample sizes exceeding 2,000, confirming strong convergence and well-mixed chains. The Bayesian R² was approximately 0.13, indicating that age, BMI, sex, and race collectively explained about 13% of variability in diabetes risk at the population level.

Model comparison results using leave-one-out cross-validation are presented below.

# Reduced models (let brms compile if needed)
invisible(capture.output({
  fit_no_race <- update(bayes_fit, formula = update(fml_bayes, . ~ . - race4))
  fit_no_sex  <- update(bayes_fit, formula = update(fml_bayes, . ~ . - sex))
}))

# Compute LOO for all three models
loo_base    <- loo::loo(bayes_fit)
loo_no_race <- loo::loo(fit_no_race)
loo_no_sex  <- loo::loo(fit_no_sex)

# Compare (pass objects positionally)
cmp <- loo::loo_compare(loo_base, loo_no_race, loo_no_sex)

# Make row names explicit and sort best-to-worst already by loo_compare
cmp_df <- as.data.frame(cmp)
cmp_df$Model <- rownames(cmp_df)
cmp_df <- cmp_df[, c("Model", setdiff(names(cmp_df), "Model"))]

knitr::kable(
  cmp_df,
  caption = "LOO comparison (higher elpd_loo is better; table sorted best to worst)."
)

Bayesian model comparison (LOO): base model vs. models without race or without sex.

	Model	elpd_diff	se_diff	elpd_loo	se_elpd_loo	p_loo	se_p_loo	looic	se_looic
bayes_fit	bayes_fit	0.000000	0.000000	-1574.573	56.98453	8.291612	0.5425672	3149.146	113.9691
fit_no_sex	fit_no_sex	-2.014434	3.294323	-1576.587	57.14599	6.459157	0.4351583	3153.175	114.2920
fit_no_race	fit_no_race	-14.736906	6.291956	-1589.310	54.54774	5.446023	0.3593072	3178.620	109.0955

Leave-one-out (LOO) cross-validation showed that models excluding race or sex had lower expected log predictive density (elpd), indicating a poorer fit. This supports the inclusion of both variables as meaningful contributors to predictive performance and overall model adequacy.

Figures below visualize posterior distributions, MCMC diagnostics, and model fit.

adult_means <- adult_imp1 %>% summarise(
  age_mean = mean(age, na.rm = TRUE),
  age_sd   = sd(age, na.rm = TRUE),
  bmi_mean = mean(bmi, na.rm = TRUE),
  bmi_sd   = sd(bmi, na.rm = TRUE)
)

to_model_row <- function(age_raw, bmi_raw, sex_lab, race4_lab) {
  tibble(
    age_c  = (age_raw - adult_means$age_mean)/adult_means$age_sd,
    bmi_c  = (bmi_raw - adult_means$bmi_mean)/adult_means$bmi_sd,
    sex    = factor(sex_lab,   levels = levels(adult_imp1$sex)),
    race4  = factor(race4_lab, levels = levels(adult_imp1$race4)),
    wt_norm = 1
  )
}

plot_post_density <- function(df_row, title_txt) {
  phat <- posterior_linpred(bayes_fit, newdata = df_row, transform = TRUE)
  ci95 <- quantile(phat, c(0.025, 0.975))
  ggplot(data.frame(pred = as.numeric(phat)), aes(x = pred)) +
    geom_density() +
    geom_vline(xintercept = ci95[1], linetype = "dashed") +
    geom_vline(xintercept = ci95[2], linetype = "dashed") +
    coord_cartesian(xlim = c(0, 1)) +
    labs(x = "P(Diabetes = 1)", y = "Density", title = title_txt) +
    theme_minimal()
}

# Use participants from analytic data (adult) for realistic inputs
stopifnot(nrow(adult) >= 2)
p1 <- to_model_row(adult$age[1], adult$bmi[1],
                   as.character(adult$sex[1]), as.character(adult$race4[1]))
p2 <- to_model_row(adult$age[2], adult$bmi[2],
                   as.character(adult$sex[2]), as.character(adult$race4[2]))

plot_post_density(p1, "Participant 1: Posterior Predictive Distribution (95% CrI)")

plot_post_density(p2, "Participant 2: Posterior Predictive Distribution (95% CrI)")

# “New” participant example
new_pt <- to_model_row(age_raw = 45, bmi_raw = 32, sex_lab = "Female", race4_lab = "Hispanic")
plot_post_density(new_pt, "New Participant: Posterior Predictive Distribution (95% CrI)")

bayesplot::mcmc_areas(
as.array(bayes_fit),
regex_pars = "^b_",
prob = 0.95
)

Figure 1: Posterior distributions (95% credible mass) for slope parameters.

bayesplot::mcmc_trace(
as.array(bayes_fit),
regex_pars = "^b_"
)

Figure 2: Trace plots for slope parameters (chain mixing and stationarity).

post_array <- posterior::as_draws_array(bayes_fit)  # draws x chains x parameters
bayesplot::mcmc_acf(post_array, pars = c("b_age_c", "b_bmi_c"))

Figure 3: Autocorrelation plots for posterior samples of age and BMI coefficients (MCMC diagnostics).

par_names <- names(posterior::as_draws_df(bayes_fit))
want <- c("b_age_c","b_bmi_c","b_sexFemale","b_race4Black","b_race4Hispanic","b_race4Other")
have <- intersect(want, par_names)

if (length(have) == 0) {
knitr::asis_output("**No matching slope parameters found for mcmc_areas.**")
} else {
bayesplot::mcmc_areas(as.array(bayes_fit), pars = have, prob = 0.95)
}

Figure 4: Posterior density areas (95% credible mass) for age, BMI, sex, and race coefficients.

bayesplot::pp_check(bayes_fit, type = "bars", nsamples = 100)

Figure 5: Posterior predictive check: observed vs. replicated outcome distribution (bars).

yobs <- adult_imp1$diabetes_ind

yrep <- brms::posterior_predict(bayes_fit, ndraws = 400)  # draws x observations

bayesplot::ppc_stat(y = yobs, yrep = yrep, stat = "mean")
bayesplot::ppc_stat(y = yobs, yrep = yrep, stat = "sd")

Figure 6: Posterior predictive checks for mean and standard deviation of the binary outcome.

Figure 7: Posterior predictive checks for mean and standard deviation of the binary outcome.

Model Fit and Calibration

Posterior predictive checks showed that simulated outcome distributions closely matched the observed diabetes prevalence, indicating strong model calibration. Both the mean and standard deviation of replicated outcomes aligned with observed data, suggesting the model adequately captured central tendency and dispersion. These results provide graphical evidence of good fit and reinforce that the priors did not unduly constrain the posterior.

or_plot <- bayes_or %>%
  dplyr::filter(term != "Intercept") %>%
  dplyr::mutate(
    term_clean = dplyr::case_when(
      term == "age_c"            ~ "Age (per 1 SD)",
      term == "bmi_c"            ~ "BMI (per 1 SD)",
      term == "sexFemale"        ~ "Female (vs. Male)",
      term == "sexMale"          ~ "Male (vs. Female)",
      term == "race4Black"       ~ "Black (vs. White)",
      term == "race4Hispanic"    ~ "Hispanic (vs. White)",
      term == "race4Other"       ~ "Other (vs. White)",
      TRUE                       ~ term
    )
  )

ggplot(or_plot, aes(x = OR, y = reorder(term_clean, OR))) +
  geom_vline(xintercept = 1, linetype = 2) +
  geom_point() +
  geom_errorbarh(aes(xmin = LCL, xmax = UCL), height = 0.15) +
  labs(x = "Odds ratio (logit model, posterior)", y = NULL)

Figure 8: Posterior odds ratios (points) with 95% credible intervals (lines).

Calibration between predicted and observed diabetes probabilities is displayed in Figure 9.

pred_mean <- colMeans(brms::posterior_epred(bayes_fit))
ggplot(data.frame(pred = pred_mean, obs = yobs),
       aes(x = pred, y = obs)) +
  geom_point(alpha = 0.15, position = position_jitter(height = 0.03)) +
  geom_smooth(method = "loess", se = TRUE) +
  labs(x = "Mean predicted probability", y = "Observed diabetes (0/1)")

Figure 9: Observed outcome vs. mean predicted probability (calibration scatter with smoother).

post_pred <- brms::posterior_epred(bayes_fit, summary = FALSE)
post_prev <- rowMeans(post_pred)
obs_prev  <- mean(adult_imp1$diabetes_ind, na.rm = TRUE)
post_prev_mat <- matrix(post_prev, ncol = 1, dimnames = list(NULL, "Prevalence"))

p <- bayesplot::mcmc_dens(post_prev_mat)
p + ggplot2::labs(title = "Posterior diabetes prevalence",
                  x = "Predicted prevalence", y = NULL) +
    ggplot2::geom_vline(xintercept = obs_prev, linetype = 2)

p

Figure 10: Posterior predictive distribution of diabetes prevalence compared to observed NHANES prevalence.

Figure 11: Posterior predictive distribution of diabetes prevalence compared to observed NHANES prevalence.

The posterior predictive distribution of diabetes prevalence closely mirrored the survey-estimated prevalence, with the posterior mean aligning within 1% of the observed rate. This indicates that the Bayesian model accurately reproduced the population-level prevalence and supports its calibration for epidemiologic inference.

# Posterior predictive prevalence (replicated datasets)

yrep <- brms::posterior_predict(bayes_fit, ndraws = 2000)   # draws x observations (0/1)
post_prev <- rowMeans(yrep)                                 # prevalence each posterior draw

# Survey-weighted observed prevalence (population estimate)

des_obs <- survey::svydesign(
id = ~SDMVPSU, strata = ~SDMVSTRA, weights = ~WTMEC2YR,
nest = TRUE, data = adult_imp1
)
obs <- survey::svymean(~diabetes_ind, des_obs, na.rm = TRUE)
obs_prev  <- as.numeric(obs["diabetes_ind"])
obs_se    <- as.numeric(SE(obs)["diabetes_ind"])
obs_lcl   <- max(0, obs_prev - 1.96 * obs_se)
obs_ucl   <- min(1, obs_prev + 1.96 * obs_se)

# Plot: posterior density with weighted point estimate and 95% CI band

ggplot(data.frame(prev = post_prev), aes(x = prev)) +
geom_density(alpha = 0.6) +
annotate("rect", xmin = obs_lcl, xmax = obs_ucl, ymin = 0, ymax = Inf, alpha = 0.15) +
geom_vline(xintercept = obs_prev, linetype = 2) +
coord_cartesian(xlim = c(0, 1)) +
labs(x = "Diabetes prevalence", y = "Posterior density",
subtitle = sprintf("Survey-weighted NHANES prevalence = %.1f%% (95%% CI %.1f–%.1f%%)",
100*obs_prev, 100*obs_lcl, 100*obs_ucl)) +
theme_minimal()

Figure 12: Population (NHANES survey-weighted) vs posterior predictive diabetes prevalence.

src_obj <- if (exists("bayes_fit_prior") && !is.null(bayes_fit_prior)) bayes_fit_prior else bayes_fit
draws_names <- names(posterior::as_draws_df(src_obj))
sort(grep("^(b_|prior_b_)", draws_names, value = TRUE))

[1] "b_age_c"         "b_bmi_c"         "b_Intercept"     "b_race4Black"   
[5] "b_race4Hispanic" "b_race4Other"    "b_sexMale"      

src_obj <- if (exists("bayes_fit_prior") && !is.null(bayes_fit_prior)) bayes_fit_prior else bayes_fit
draws_df <- posterior::as_draws_df(src_obj)
all_cols <- names(draws_df)

# EDIT THESE if your names differ (check the printed list from list-brms-pars)

want_post <- c("b_age_c","b_bmi_c","b_sexFemale","b_race4Black","b_race4Hispanic","b_race4Other")

have_post  <- intersect(want_post, all_cols)
have_prior <- intersect(paste0("prior_", have_post), all_cols)

pairs <- data.frame(post = have_post, prior = paste0("prior_", have_post),
stringsAsFactors = FALSE)
pairs <- pairs[pairs$prior %in% have_prior, , drop = FALSE]

if (nrow(pairs) == 0) {
knitr::asis_output("**No matching prior/posterior parameters found to overlay.**")
} else {
post_long <- tidyr::pivot_longer(
draws_df[, pairs$post, drop = FALSE],
cols = tidyselect::everything(), names_to = "term", values_to = "estimate"
)
post_long$type <- "Posterior"

prior_tmp <- draws_df[, pairs$prior, drop = FALSE]
colnames(prior_tmp) <- pairs$post
prior_long <- tidyr::pivot_longer(
prior_tmp,
cols = tidyselect::everything(), names_to = "term", values_to = "estimate"
)
prior_long$type <- "Prior"

combined_draws <<- dplyr::bind_rows(prior_long, post_long)

lbl <- c(
b_age_c = "Age (1 SD)", b_bmi_c = "BMI (1 SD)",
b_sexFemale = "Female vs Male",
b_race4Black = "Black vs White",
b_race4Hispanic = "Hispanic vs White",
b_race4Other = "Other vs White"
)
combined_draws$term <- factor(
combined_draws$term,
levels = intersect(names(lbl), unique(combined_draws$term)),
labels = lbl[intersect(names(lbl), unique(combined_draws$term))]
)

ggplot2::ggplot(combined_draws, ggplot2::aes(x = estimate, linetype = type)) +
ggplot2::geom_density() +
ggplot2::facet_wrap(~ term, scales = "free", ncol = 2) +
ggplot2::labs(x = "Coefficient (log-odds)", y = "Density", linetype = NULL) +
ggplot2::theme_minimal()
}

No matching prior/posterior parameters found to overlay.

Figure 13: Prior (dashed) vs posterior (solid) densities for selected coefficients.

if (exists("combined_draws") && is.data.frame(combined_draws) && nrow(combined_draws) > 0) {
ggplot(combined_draws, aes(x = estimate, fill = type)) +
geom_density(alpha = 0.4) +
facet_wrap(~ term, scales = "free", ncol = 2) +
theme_minimal(base_size = 13) +
labs(
title = "Prior vs Posterior Distributions",
x = "Coefficient estimate",
y = "Density",
fill = ""
)
} else {
knitr::asis_output("**Skipped: no matching prior/posterior draws to plot.**")
}

Skipped: no matching prior/posterior draws to plot.

Figure 14: Prior vs Posterior Distributions (ggplot2 version).

par_names <- names(posterior::as_draws_df(bayes_fit))
want <- c("b_age_c","b_bmi_c","b_sexFemale","b_race4Black","b_race4Hispanic","b_race4Other")
have <- intersect(want, par_names)

if (length(have)) {
  bayesplot::mcmc_areas(as.array(bayes_fit), pars = have, prob = 0.95)
} else {
  knitr::asis_output("**No matching slope parameters available to plot.**")
}

Figure 15

For age and BMI, the posterior densities shift notably away from the N(0, 2.5) prior toward positive values and are narrower, indicating strong information from the data; for sex, the posterior remains closer to the prior with more overlap, indicating weaker evidence.

The overlay of prior and posterior densities illustrates that informative updates occurred primarily for BMI, age, and race coefficients, which showed distinct posterior shifts relative to the priors. In contrast, weaker predictors such as sex displayed overlapping distributions, indicating that inference for those parameters was more influenced by prior uncertainty than by the observed data. This balance confirms appropriate regularization rather than overfitting.

Results

A concise summary of posterior estimates is provided below.

cat(paste(bullets, collapse = "\n"))

Population-level interpretation (posterior, odds ratios)

• Convergence. All R-hat ≈ 1.00; large ESS → excellent mixing.
• Baseline risk. Male, White, mean age/BMI: ~5.2% predicted diabetes prevalence.
• Age. +1 SD → 2.93× (95% CrI 2.62–3.30; CrI excludes 1).
• BMI. +1 SD → 1.92× (95% CrI 1.76–2.10; CrI excludes 1).
• Female vs. Male. NA× (95% CrI NA–NA; CrI overlaps 1).
• Black vs. White. 1.62× (95% CrI 1.23–2.12; CrI excludes 1).
• Hispanic vs. White. 1.82× (95% CrI 1.40–2.38; CrI excludes 1).
• Other/Multi vs. White. 2.11× (95% CrI 1.46–2.99; CrI excludes 1).

# Combine results from all three models

svy_tbl   <- if (exists("svy_or") && nrow(svy_or) > 0)
dplyr::mutate(svy_or,   Model = "Survey-weighted (MLE)") else NULL
mi_tbl    <- if (exists("mi_or") && nrow(mi_or) > 0)
dplyr::mutate(mi_or,    Model = "MICE Pooled") else NULL
bayes_tbl <- if (exists("bayes_or") && nrow(bayes_or) > 0)
dplyr::mutate(bayes_or, Model = "Bayesian") %>%
dplyr::filter(term != "Intercept") else NULL

all_tbl <- dplyr::bind_rows(Filter(Negate(is.null), list(svy_tbl, mi_tbl, bayes_tbl))) %>%
dplyr::mutate(
term = dplyr::case_when(
  grepl("^bmi", term,  ignore.case = TRUE) ~ "BMI (per 1 SD)",
  grepl("^age", term,  ignore.case = TRUE) ~ "Age (per 1 SD)",
  grepl("^sexFemale$", term)               ~ "Female (vs. Male)",
  grepl("^sexMale$", term)                 ~ "Male (vs. Female)",
  grepl("^race4Hispanic$", term)           ~ "Hispanic (vs. White)",
  grepl("^race4Black$", term)              ~ "Black (vs. White)",
  grepl("^race4Other$", term)              ~ "Other (vs. White)",
  TRUE ~ term
),
OR_CI = sprintf("%.2f (%.2f – %.2f)", OR, LCL, UCL)
) %>%
dplyr::select(Model, term, OR_CI)

Comparative odds ratios across frameworks are shown in Table 7.

knitr::kable(all_tbl, align = c("l","l","c"))

Table 7: Comparison of odds ratios (per 1 SD for age and BMI) and 95% intervals across survey-weighted, MICE, and Bayesian frameworks.

Model	term	OR_CI
Survey-weighted (MLE)	Age (per 1 SD)	2.98 (2.70 – 3.28)
Survey-weighted (MLE)	BMI (per 1 SD)	1.93 (1.68 – 2.22)
Survey-weighted (MLE)	Male (vs. Female)	1.29 (1.02 – 1.63)
Survey-weighted (MLE)	Hispanic (vs. White)	1.81 (1.43 – 2.29)
Survey-weighted (MLE)	Black (vs. White)	1.60 (1.16 – 2.21)
Survey-weighted (MLE)	Other (vs. White)	2.20 (1.46 – 3.30)
MICE Pooled	scale(age)	2.90 (2.61 – 3.21)
MICE Pooled	scale(bmi)	1.81 (1.66 – 1.96)
MICE Pooled	relevel(sex, “Male”)Female	0.81 (0.68 – 0.96)
MICE Pooled	relevel(race4, “White”)Hispanic	2.07 (1.66 – 2.59)
MICE Pooled	relevel(race4, “White”)Black	1.79 (1.43 – 2.24)
MICE Pooled	relevel(race4, “White”)Other	2.00 (1.51 – 2.66)
Bayesian	Age (per 1 SD)	2.93 (2.62 – 3.30)
Bayesian	BMI (per 1 SD)	1.92 (1.76 – 2.10)
Bayesian	Male (vs. Female)	1.28 (1.06 – 1.55)
Bayesian	Hispanic (vs. White)	1.82 (1.40 – 2.38)
Bayesian	Black (vs. White)	1.62 (1.23 – 2.12)
Bayesian	Other (vs. White)	2.11 (1.46 – 2.99)

This table summarizes results from the survey-weighted (design-based), multiple-imputation, and Bayesian models.

The Bayesian model’s credible intervals closely align with the frequentist confidence intervals but provide a more direct probabilistic interpretation of uncertainty.

Across all three frameworks—survey-weighted (MLE), multiple imputation, and Bayesian—age and BMI were consistently associated with higher odds of doctor-diagnosed diabetes. Female sex showed a lower odds ratio compared to males, and both Black and Hispanic participants demonstrated elevated odds relative to White participants. The similarity of effect sizes across frameworks underscores the robustness of these predictors to different modeling assumptions and missing-data treatments. Bayesian credible intervals largely overlapped frequentist confidence intervals, confirming stability of inference while allowing direct probabilistic interpretation.

Discussion and Limitations

Interpretation

The Bayesian logistic regression framework produced results that were highly consistent with both the survey-weighted and MICE-pooled frequentist models. Age and BMI remained the most influential predictors of doctor-diagnosed diabetes, each showing a strong and positive association with diabetes risk.

Unlike classical maximum likelihood estimation, the Bayesian approach directly quantified uncertainty through posterior distributions, offering richer interpretability and more transparent probability statements. The alignment between Bayesian and design-based estimates supports the robustness of these associations and highlights the practicality of Bayesian modeling for complex, weighted population data.

Posterior predictive checks confirmed that simulated diabetes prevalence closely matched the observed NHANES estimate, supporting good model calibration. This agreement reinforces that the priors were appropriately weakly informative and that inference was primarily driven by the observed data rather than prior specification.

Overall, this study demonstrates that Bayesian inference complements traditional epidemiologic methods by maintaining interpretability while enhancing stability and explicitly quantifying uncertainty in complex survey data. These consistent findings across modeling frameworks underscore the robustness of core risk factors and support the use of Bayesian inference for epidemiologic research involving complex survey data.

Limitations

While this analysis demonstrates the value of Bayesian logistic regression for epidemiologic modeling, several limitations should be acknowledged.

First, the use of a single imputed dataset for the Bayesian model—rather than full joint modeling of imputation uncertainty—may understate total variance.

Second, NHANES exam weights were normalized and treated as importance weights, which approximate but do not fully reproduce design-based inference.

Third, the weakly informative priors \(N(0, 2.5)\) for slopes and Student-t(3, 0, 10) for the intercept were not empirically tuned; alternative prior specifications could slightly alter posterior intervals.

Finally, although convergence diagnostics (R̂ ≈ 1, sufficient effective sample sizes, and stable posterior predictive checks) indicated good model performance, results are conditional on the 2013–2014 NHANES cycle and may not generalize to later datasets or longitudinal analyses.

Conclusion

The Bayesian, survey-weighted, and imputed logistic regression frameworks all identified consistent predictors of diabetes risk in U.S. adults: advancing age, higher BMI, sex (lower odds for females), and non-White race/ethnicity.

The Bayesian model produced estimates nearly identical in direction and magnitude to the frequentist results while providing a more comprehensive assessment of uncertainty through posterior distributions and credible intervals.

These consistent findings across modeling frameworks underscore the robustness of core risk factors and support the use of Bayesian inference for epidemiologic research involving complex survey data.

By incorporating prior information and using MCMC to sample from the full posterior distribution, Bayesian inference enhances model transparency and interpretability.

Its agreement with traditional approaches underscores that Bayesian methods can be applied confidently in large-scale public health datasets such as NHANES.

Future extensions could integrate hierarchical priors, multiple NHANES cycles, or Bayesian model averaging to better capture population heterogeneity, temporal trends, and evolving diabetes risk patterns.

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